## Abstract

A Monte Carlo model was used to analyze the propagation of polarized light in linearly birefringent turbid media, such as fibrous tissues. Linearly and circularly polarized light sources were used to demonstrate the change of polarizations in turbid media with different birefringent parameters. Videos of spatially distributed polarization states of light backscattered from or propagating in birefringent media are presented.

© Optical Society of America

## 1. Introduction

Recently, there has been increased interest in the propagation of polarized light in
randomly scattering media because of its potential applications, especially in
noninvasive optical diagnosis. In recent theoretical research, Ambirajan and Look [1] used a Monte Carlo technique to study the degree of
polarization (DOP) of light diffusely reflected from turbid media. Rakovic
*et al*. [2] and Bartel *et al*. [3] developed Monte Carlo algorithms to study the backscattered
intensity patterns and compared their simulation results with experimental data. Yao
and Wang [4] used a time-resolved Monte Carlo technique to present the
propagation of polarized light in turbid media. Until now, all of these theoretical
studies are limited to isotropic turbid media, which does not possess birefringence.

Birefringence is, however, prevalent in biological tissues, such as collagen or muscle. Tissue birefringence results primarily from the linear anisotropy of the fibrous structures that form the extra-cellular background medium. The refractive index of the background medium is higher along the length of fibers than that along the cross section. In birefringent turbid media, both birefringence and scattering alters the polarization of light. Consequently, the birefringent effects of the background medium provide extraordinary information for subsurface imaging, which can be obtained by analyzing the diffusely reflected light [5–8]. This paper describes our study on the propagation of polarized light in birefringent turbid media based on a time-resolved Monte Carlo algorithm for the first time.

## 2. Monte Carlo algorithm for birefringent turbid media

Monte Carlo simulation to describe the multiple scattering events of photons in turbid media is based on radiative theory, which assumes that the scattering event is independent and has no coherence effects. The linearly birefringent tissue is looked on as a uniaxial material with the slow axis along the length of the fibers. The geometry of multiple scattering events in a birefringent turbid medium is shown in Fig. 1.

A photon is scattered in the background medium, which is anisotropic with the linear
birefringent effect, by spherical particles then exits the medium from the surface
of the sample. The cyan horizontal lines demonstrate the direction of the slow axis
of the medium background, which is along the x-axis. The basic Stokes-Mueller
formalism for tracing the propagation of each polarized photon packet in turbid
media has been described in Refs. [2–4]. Considering the birefringence of the media, we express the
Stokes vector of a diffusely reflected photon packet after it has been scattered
*n* times in a birefringent turbid medium as

$$\times \mathbf{T}({\Delta}_{2},{\beta}_{2})\mathbf{M}\left({\Theta}_{2}\right)\mathbf{R}\left({\varphi}_{12}\right)\mathbf{T}({\Delta}_{1},{\beta}_{1})\mathbf{M}\left({\Theta}_{1}\right)\mathbf{R}\left({\varphi}_{1}\right)\mathbf{T}({\Delta}_{0},{\beta}_{0}){\mathbf{S}}_{0},$$

where *µ*_{s}
,
*µ*_{a}
are the scattering and absorption
coefficients, respectively; *δ* is the linear
birefringence value; (*x*′,
*y*′) is the detection point on the upper surface of the
turbid medium in the laboratory coordinate. S_{0} and ${\mathrm{S}}_{n}^{\text{bs}}$
represent the Stokes vectors of the incident and the backscattered
photons, respectively.
[*µ*_{s}
/(*µ*_{a}
+*µ*_{s}
)]^{n}
expresses the remaining energy after the photon has been scattered
*n* times. **R**(*ϕ*) is the
rotation matrix that connects the two Stokes vectors that describe the same
polarization state but with respect to the two reference planes such that one
reference plane coincides with the other after a counterclockwise rotation by angle
*ϕ* around the direction of light propagation [2]. **M**(Θ) is the matrix for each single
scattering event based on Mie theory [9], where Θ is the scattering angle.
**T**(Δ, *β*) describes the
birefringent effect on the photon packet in each free path, which can be expressed
as

$${C}_{2}=\mathrm{cos}\left(2\beta \right),{C}_{4}=\mathrm{cos}\left(4\beta \right),{S}_{2}=\mathrm{sin}\left(2\beta \right),{S}_{4}=\mathrm{sin}\left(4\beta \right).$$

*β* is the azimuthal angle of the birefringent slow axis at
the *x-y* plane in the local coordinate of the propagating photon.
Δ is the phase retardation, which can be obtained by

where *s* is the step length; *λ* is the
wavelength of the light *in vacuo*; and
Δ*n* is the difference between the maximum and minimum
refractive indices in the plane perpendicular to the propagation orientation of the
photon packet. When the angle *α* between the slow axis of
the medium and the propagation orientation of the photon packet is known,
Δ*n* can be shown as

*n*_{s}
and *n*_{f}
are the refractive
indices of the birefringent medium along the slow axis and the fast axis,
respectively; the linear birefringence value *δ* is
*n*_{s}*-n*_{f}
.

## 3. Results

For comparison, we kept most parameters of the light and the turbid media constant in
following simulations and altered only the polarization states of the incident light
and the parameters of birefringence. *λ* is 594 nm. The
radius of the spherical scattering particles in the media is 350 nm. The refractive
index of the scattering particles is 1.57. *n*_{s}
is 1.33.
The anisotropic factor g is calculated to be 0.9.
*µ*_{s}
and
*µ*_{a}
are 90 cm^{-1} and 1
cm^{-1}, respectively.

#### 3.1 Polarization of light backscattered from birefringent turbid media

The DOLP (degree of linear polarization) patterns of backscattered light from
isotropic and birefringent turbid media for linear incident polarization are
shown in Fig. 2. When the incident light is right-circularly
polarized, the comparison of the DOCP (degree of circular polarization) patterns
is shown in Fig. 3. For both figures, *δ*
is 1.33/1000. Every picture in Figs. 2 and 3 is centered at the point of incidence. The actual size
of each picture is 1×1 mm^{2}.

When light propagates in the birefringent turbid media, the change of polarization depends on the orientations and the lengths of the discontinuous paths of the photon as well as the orientation of the birefringence in the medium, which has been shown in Eqs. 2–4. The statistic distribution of the changes of polarization in the turbid medium can be observed from the spatial polarization patterns of the backscattered light. The symmetries in the patterns can be explained by the symmetries in the single scattering matrix, the orientation of birefringence and the polarization state of the incident light. For instance, in the birefringent turbid media, the circularly polarized light no longer product a uniform distribution in the dimension of the azimuthal angle in the single scattering event; therefore, for circularly polarized incident light, the polarization patterns of the backscattered light do not have to be rotationally symmetric but depend on the orientation of the birefringence in the medium.

It is noticeable that when the light is circularly polarized and the orientation
of the birefringent slow axis is along the *z*-axis, detected
DOCP of backscattered light drops with oscillation when the source-detector
distance increases (Fig. 4). The oscillation is correlated with the periodic
phase retardation caused by the birefringence.

Figures 5(a) shows the time-resolved DOLP of
backscattered light from the birefringent turbid media with the birefringent
slow axes oriented along the 45° in the *x-y* plane,
where the incident light is linearly polarized along the
*x*-axis. The linear birefringence value
*δ* is set to be 1.33/1000 and 1.33×2/1000,
respectively. For right-circularly polarized incident light, the time-resolved
DOCP of backscattered light from the birefringent turbid media with the
birefringent slow axes oriented along the cross section of the z-axis is shown
in Fig. 5(b). The period of the oscillations in Fig. 5 is inversely proportional to δ. In
isotropic scattering media, the DOLP and DOCP of backscattered light decrease
when the time of propagation increases, because multiply scattered photons
usually have greater path length than weakly scattered photons. However, in
birefringent scattering media, the DOLP and DOCP present oscillations as a
function of time because of the periodic phase retardation caused by the
birefringence, a similar phenomenon was observed in optical-coherence tomography [10].

Videos of the polarizations of the backscattered light from an isotropic turbid
medium and birefringent turbid media with different orientations of slow axis
are shown in Figs. 6 and 7 for linear and circular incident polarizations,
respectively. The propagation time of photons changes from 0 to 3.3 ps. The
actual size of each video sequence is 1×1 mm^{2}. These
videos demonstrate how the polarization states of the diffusely reflected light
depend on the propagation time as well as the birefringence in the turbid media.

#### 3.2 Polarization propagation in single-layer birefringent turbid media

In order to demonstrate the linearly birefringent effect on polarized light in
detail, we present videos of polarization propagations in birefringent turbid
media with different orientations of slow axis, as shown in Figs. 8 and 9 for linear or circular incident polarizations,
respectively. Fig. 8(a) and Fig. 9(a) show the results for an isotropic medium for
comparison. These two-dimensional sequences represent the three-dimensional
distributions of the polarizations projected onto the *x-z* plane
in the laboratory coordinate. The actual size of each video sequence is
1×2 mm^{2}. The polarization states of the photons at the
expanding edges are retained well in the isotropic turbid medium because the
photons there experience very few scattering events. However, in the
birefringent turbid media, the polarization states at the expanding edges depend
on the geometric positions of the edges in the media, the orientation of the
birefringence in the media and the incident polarization state, though the DOP
of the light at the expanding edges is retained well. The alternate and periodic
changes between the circular and linear polarization states of the light at the
expanding edges are caused by the periodic phase retardation in the birefringent
media.

#### 3.3 Polarization propagation in two-layer birefringent turbid media

In actual biological tissues, for example human skin, the orientations of the
fibrous structures vary in different layers. In this case, the propagation of
polarized light is much more complicated, because the birefringence with
different orientations and *δ* in each layer has a
combined effect on the polarization propagation of light in the medium.

We studied the polarization propagation in a two-layer turbid medium. The
orientations of the birefringent slow axes in the upper and lower layer are
along the *y*-axis and *x*-axis, respectively. The
two layers have the same birefringent value *δ* that
equals 1.33/1000. The thicknesses of the upper and lower layers are 0.4 mm and
0.6 mm, respectively. Videos of DOP, DOLP and DOCP in this two-layer
birefringent turbid medium for different incident polarizations (the linear
polarization oriented at x-axis and the right-circular polarization) are shown
in Figs. 10 and 11, respectively. The gray dot lines in pictures show the
interface of the two layers. The actual size of each video sequence is
1×2 mm^{2}. It is noticeable that the polarization patterns
are different at the upper and lower sides of the two-layer interface because of
the different orientations of the birefringence in the two layers.

## 4. Conclusion

We theoretically studied the propagations of polarized light in birefringent turbid
media based on a time-resolved Monte Carlo algorithm. The polarization patterns of
backscattered light and the spatially distributed polarization states in the
birefringent turbid media were analyzed. Linearly polarized light and circularly
polarized light were considered as sources. Results for media with different
birefringent parameters including the orientation of fibers and the birefringence
value *δ* were compared and discussed.

The theoretic study presented in this paper should contribute to the understanding of the essential physical processes of polarized light propagation in birefringent tissues. The Monte Carlo simulation cannot model coherent phenomena of tissue optics, however this method can be applied in the non-coherent regime or in cases where coherent effects are removed. Some phenomena observed in the simulation can provide potential methods for use in actual subsurface tomography of fibrous tissues.

Acknowledgments This project was sponsored in part by National Institutes of Health grants R01 CA71980 and R21 RR15368, National Science Foundation grant BES9734491, and Texas Higher Education Coordinating Board grant 000512-0123-1999.

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